Q & A
Q.10 It is appreciable. Could you next explain the meaning
of nature of mathematics?
Mathematics reveals hidden patterns that helps us to
understand the world around us. Much more than
arithmetic and geometry, Mathematics today is a diversified
discipline, which deals with data, measurement and
observations from science; with reference, deduction and
proof, and with mathematical models, natural phenomena,
human behaviour and social systems. As a practical matter,
mathematics is a science of patterns and order. Its domain
is not molecules or cells, but numbers, chance, form,
algorithm and change. As a science of abstract object,
mathematics relies on logic rather than on observation as
its standard of truth, yet employs observation, simulations
and even experimentation as means of discovering the truth.
The result of mathematics—theorems and theories—are
both significant and useful; the best results are also elegant
and deep. In addition to theorems and theories,
mathematics offers distinctive mode of thought which are
both versatile and powerful, including mathematical
modelling, abstraction, optimisation, logical analysis,
inference from data and use of symbols. Due to diverse
application of mathematics, the various mathematical tools
are required which are interlinked with each other. It is the
tall shape of mathematics.
Q.11 What is the meaning of ‘the tall shape of mathematics’?
Many concepts are needed to be learnt sequentially in
mathematics. Only after mastering arithmetic, algebra is
learnt, and only when one can factor polynomials, is able
to understand trigonometry, and so on. Thus, since each
theme is built on another, it results in a tall shape. This
makes it difficult for children; someone who finds one stage
difficult finds it hard to catch up later.
Q.12 But, as I understand, that is the nature of
mathematics. What does NCF-05 say about it?
NCF-2005 says that the tall shape of mathematics can be
de-emphasised in favour of a broad-based curriculum
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with more topics that start from the basics. Revisiting the
basics of mathematics at secondary and higher secondary
stages will help children make better use of their time
Q.13 We often face a difficult choice, especially at the
secondary and higher secondary stages, of deciding
whether we should teach many topics without much
detail, giving children exposure to those topics, or
should we cover a few themes in depth, giving children
competence. What is the solution to this problem?
There are arguments in favour of either choice. It is
generally not possible to do both, since there are often
conflicting demands of depth versus breadth. There is no
general answer to this question. The teacher is the best
person to find the right balance, in the given local situation
Q.14 I appreciate that NCF-2005 advocates this flexibility.
I would like to know, whether the meaning of
‘constructivism’ is the same in the context of
mathematics as it is in science?
It means the same thing; an approach by which children
discover and construct their knowledge, rather than it
being simply given and taken uncritically. In mathematics,
for example, this means that children’s ability to come up
with a formula is more important than being able to
correctly use well known formulae.
Q.15 I understand. It means that discovering even simple
facts (theorems) on their own, and arguing why they
are true is more important than being able to recall
famous theorems and their proofs. Am I right?
Absolutely. Children view mathematics as something to
talk about, to communicate, to discuss among
themselves, to work together on. Making mathematics a
part of children’s life experience is the best mathematics
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Q.16 I try my best to help children to discover the formulae
on their own and I have observed that this way they
enjoy mathematics rather than fear it. I understand
that mathematics is more than formulae and
A.16 That is great.
Q.17 What is the meaning of the term ‘multiplicity of
Very often, there are many ways of solving a problem, many
procedures for computing a quantity, many ways of proving
or presenting an argument. Offering such a choice allows
children to work out and use the approach that is most
natural and easy for them. For some students who learn
more than one approach, this is a technique for self-
checking. Multiplicity of approaches is crucial for liberating
school mathematics from the tyranny of the one right
answer, found by applying the one algorithm taught. When
many ways are available, one can compare them, decide
which is appropriate when, and in the process gains insight.
For instance, to subtract 53 from 100, you could use
the standard algorithm of taking away with borrowing, or
consider how people do this in shops. When someone buys
material for Rs 53 and gives a hundred-rupee note, the
shopkeeper may return as follows: here, four notes of rupees
10, another five rupees and two coins of rupee 1. Here even
the answer, 47, is not mentioned but the operation is
correct. (This is not an argument to say that children need
not learn the standard method, but to say that for children
having difficulties, alternatives may help, until they gain
Q.18 What does it mean to shift focus from content to
In mathematics, content areas are well established:
arithmetic, algebra, geometry, trigonometry, mensuration,
etc. Our teaching is content oriented, and while it is
important to teach content, it is even more important to think
of how we teach such content. Here process refers to
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pedagogic techniques. For example, many general tactics
of problem solving can be taught progressively during the
different stages of school. Techniques like abstraction,
quantification, analogy, case analysis, reduction to simple
situations, even guess-and-verify, are useful in many
problem contexts. When children learn a variety of
approaches (over a period of time), their tool-kit gets richer
and, as we talked about it above, they also learn which
approach is best suitable in a given situation. Instead of
looking at whether children know something, it is more
important to observe how they acquire such knowledge.
Though the processes cut across several subject areas, these
are central to mathematics. Problem solving, estimation of
quantities, approximating solutions, visualisation and
representation and mathematical communications are some
of the processes of mathematics. As an example, an ability
to convert grams into kilograms is important, but more
important is the capacity to talk in terms of kilograms for
weight of cabbage, and grams for eggs.
Q.19 What is the meaning of mathematical communication?
Precise and unambiguous use of language and rigour in
formulation are important characteristics of mathematical
treatment. The use of jargon in mathematics is deliberate
and stylised. Discussing with appropriate notations aids
thinking. That is what mathematical communication
Q.20 I ensure that children give as much importance to
setting up the equations as to solving them. Is that
also mathematical communication?
A.20 Yes, you do the right thing.
Q.21 What is the difference between ‘word problem and
In word problem, we do not care for physical insight of the
problem, but in mathematical modelling, physical insight
of the problem is more important. The term modelling is
typically used at the secondary stage and later, for
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situations where students come up with a mathematical
(typically algebraic) formulation to solve them, translating
the answers back into the situation. The model is intended
to be used later on for other similar purposes. Word
problems are similar, but typically used at the elementary
stage, and refer to exercises where the child formalises the
situation into a form where a specific mathematical
technique can be applied. We can think of word problems
as ‘disposable’ (or ‘use and throw’) models!
Q.22 Distinguish between use of concrete models,
mathematical models and mathematical modelling.
When we talk of using concrete models, we are referring to
models already built by self or others which make it simpler
to comprehend a difficult concept, and to visualise it; for
example, cone, cylinder, frustum of cone, etc. used in
mathematics laboratory. The meaning of mathematical
model is to connect physical situation into mathematics
with the help of symbols, such as calculating simple interest
with the help of the formula
, where symbols
have their usual meanings.
Mathematical modelling, on the other hand, is a
process of transformation of a physical situation into
mathematical analogies with appropriate conditions. It
may be an iterative process where we start from a crude
model and gradually refine it until it is suitable for solving
the problem and enables us to gain insight into and
understanding of the original situation. For example,
constructing a mathematical model for the estimation of
number of fish in a pond without accessing the situation,
estimating the number of trees in a dense forest, etc.
Q.23 What is systematic study of space and shapes?
Space is all around us, and we see shapes all the time.
But when we see a half-filled glass of water, only
systematic study opens our eyes to the circular base,
the cylindrical body, an estimate of the volume of water,
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etc. Similarly, geometry gives us a sense of symmetry
and stability when we look at architecture. Thus,
geometry, its systematic understanding using
quantities, and its principles (theorems) altogether can
be considered the beginning of systematic study of
space and shapes.
Q.24 What is tyranny of procedure?
When we learn to do something only as a procedure: “do
this, then that, then this”, without understanding why, we
not only make mistakes, but also become incapable of
applying this learning in a slightly changed situation. A
good cook knows not merely the recipe, but also the role-
played by each ingredient, so that she can use a different
one when something is not available. In mathematics,
simply learning formulae without understanding makes
for such tyranny, as a consequence, we stop thinking.
Q.25 What is meaningful problem solving?
Exercises at the end of a chapter typically involve only
application of a specific skill learnt in a specific situation.
Problem solving is more general and should be
distinguished. The problem-solving situation is meaningful,
when it interests the child who is then motivated to solve
the problem, the situation is genuine and the solution is
relevant. Meaningfulness is different from interesting
stories. A problem asked: Mother made 120 puris, 5 people
ate 22 each, how many were left over ? This is utterly
meaningless! Even if one can produce a family with 5 people
eating exactly the same number of puris, who cares how
many are left over after each one ate 22 of them? The only
purpose of asking such a question may be to have children
reflect on the atrocity of having their mother (or anyone
else for that matter) to make 120 puris in the first place!
On the other hand, finding the number of bricks required
to make a wall of given dimension is a meaningful problem
as it involves the use of concepts of volume as well as
concepts of division.
Q & A
Q.26 Why is problem posing as important as problem
Problem solving usually means a better understanding of
the concepts involved, which in turn helps in solving the
problem. Problem posing, on the other hand, often requires
original and diverse thinking and has many a time
resulted in the development of mathematics. For example;
(i) Many attempts were made to prove/disprove Euclid’s
postulate. The work done during these attempts
resulted in the development of Non-Euclidean Geometry;
(ii) Famous seven bridges problem resulted in the
development of a new branch of mathematics called Graph
Q.27 What is visual learning in mathematics and why is it
Learning mathematics by drawing visuals such as number
line, bar graph, line graph, histogram, pie chart, etc., is
called visual learning. It is important because it facilitates
learning, makes it more permanent, and facilitates
communication of ideas or result.
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